Solution using conditional probability: we have seen that if seat $1$ is chosen before the last passenger is seated, then that passenger sits in their assigned seat; if seat $N$ is chosen before the last passenger is seated, then the last passenger is displaced and must take seat $1$. Let $E_k$ be the event that the $k$th passenger to be seated is the first to take one of seats $1$ and $N$. Then $$ \Pr(\text{$N$ gets their assigned seat})=\sum_{k=1}^{N-1}\Pr(\text{$k$ sits in $1$}\mid E_k)\Pr(E_k)=\frac{1}{2}\sum_{k=1}^{N-1}\Pr(E_k)=\frac{1}{2}. $$ Diagrammatically, this calculation is shown below. The probability that $N$ gets their assigned seat is the sum of the probabilities that oneeach of passengers $1$ through $N-1$ takes seat $1$.
Arctic Char
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Will Orrick forgot to link to that poster's answer.
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